The practice of Ellipsometry provides a method to characterize samples which have at least one surface layer thereupon. Values for thickness and optical constants are determined by obtaining ellipsometric data, proposing a mathematical model of the sample and performing a comparison procedure, such as mathematical regression, to evaluate parameters in said mathematical model. Mathematical regression, however, works best when good starting values for parameters in the mathematical model are utilized. This is particularly true for parameters which are more significant than others in the model. One approach to determining starting values is to provide a grid of values and test numerous combinations. This provides utility, but can be tedious and time consuming.
Patents that describe obtaining good initial values are:    U.S. Pat. No. 5,889,592 to Zawaideh which describes simultaneously measuring optical constants and thickness of single and multilayer films. Discussed are use of “Global Optimization Algorithms” to minimize error.    U.S. Pat. No. 4,999,509 to Wada et al. describes the use of global and local optimization methods to determine film thickness of single and multilayer films.    U.S. Pat. No. 5,953,446 to Opsal et al. describes use of genetic algorithms.    U.S. Pat. No. 6,278,519 to Rosenscwaig et al., describes common routines/algorithms used in optical measurements and genetic algorithms to search parameter space.
Papers that concern the topic are:    “Multiple Minima in the Ellisometric Error Function”, Alterovitz & Johs, Thin Solid Films, (1998), which describes how to find correct global minima which multiple local minima are present. A Grid Search is described which involves searching a range of possible values for parameters and combinations thereof and results are analyzed by the Levenberg-Marquardt algorithm to identify the smallest Error Function (MSE).    “Simultaneous Measurement of Six layers in a Silicon Insulator Film Stack Using Spectrophotometry and Beam Profile Reflectometry”, Lang et al., J. App. Phys., 81(8) (1997). This paper describes a global optimization for a system of up to 12 parameters. The Levenberg-Marquardt algorithm is applied and a best solution is found as a global minimum. A Cauchy dispersion model is used to describe optical constants.    “Parameter Correlation and Precision in Multiple-Angle Ellipsometry”, Bu-Abbud & Bashara, Applied Optics 20 (1981). This article describes solutions for desired variables/parameters when many angles of incidence are used. A two step approach is used of doing a Global Search to obtain First Estimates of Parameters, followed by a Detailed Regression.    “Overview of Variable Angle Spectroscopic Ellipsometry (VASE), Part 1: Basic Theory and Typical Applications”, Woollam et al. SPIE Proc. CR72 (1999). This article describes use of the Levenberg-Marquardt algorithm along with global-searching to find the correct “global” minimum.    “Overview of Variable Angle Spectroscopic Ellipsometry (VASE), Part 1: Basic Theory and Typical Applications”, Woollam et al. SPIE Proc CR72 (1999). This paper discusses detail optical constant parameterization using dispersion models.    “Effective Dielectric Function of Mixtures of Three or More Materials: a Numerical Procedure for Computation”, Bosch et al., Surface Science 453 (2000). This article describes applying minimization algorithm techniques applied in Effective Medium Approximation equations.    “Characterization of Multilayer GaAs/AlGaAs Transistor Structures by Variable Angle Spectroscopic Ellipsometry”, Merkle et al., Jap. J. of Appl. Phys. 28, (1989). This article describes spanning a large range of parameter values to arrive at good-fit parameters.    “Spectroellipsometric Characterization in Inhomogeneous Films”, Tirri et al., SPIE Proc. 794 (1987). This article describes use of global optimization algorithms to find correct “global” minimum. Sensitivity factors and model results are provided.    “High Precision UV-Visible-Near-IR Stokes Vector Spectroscopy”, Zettler et al., Thin Solid Films, 234 (1993). This article describes KK consistency applied to Cubic Splines.
Books identified are:    “Numerical Mathematics and Computing” by Cheney and Kincaid, Third Edition, Brooks/Cole Publishing Company, 1994, which describes Splines, and is incorporated by reference herein; and    “Techniques for Characterization of Electrodes and Electrochemical Processes”, Varma & Selman, John Wiley & Sons, Chp. 2 “Ellipsometry as in Situ Probe for the Study of Electrode Processes” Mueller.
Even in view of the known prior art, need exists for improved methodology for identifying good starting values for mathematical model equation coefficients when a square error reduction fit is performed to fit the mathematical model to data.